On Semiclassical Limits of String States
aa r X i v : . [ h e p - t h ] J un YITP-SB-07-01
On Semiclassical Limits of String States
Jose J. Blanco-Pillado a, , Alberto Iglesias b, and Warren Siegel c, a Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, MA 02155 b Department of Physics, University of California, Davis, CA 95616 c C. N. Yang Institute for Theoretical Physics, State University of New York, Stony Brook, NY 11794
Abstract
We explore the relation between classical and quantum states in both open andclosed (super)strings discussing the relevance of coherent states as a semiclassicalapproximation. For the closed string sector a gauge-fixing of the residual world-sheetrigid translation symmetry of the light-cone gauge is needed for the construction tobe possible. The circular target-space loop example is worked out explicitly. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]
Introduction
The motivation to investigate the semiclassical limit of fundamental strings is twofold.First, in view of the revived interest in the possibility of producing superstrings of cosmicsize in models of brane inflation, it has been suggested that brane annihilation wouldleave behind a network of lower dimensional extended objects [1, 2, 3, 4] which wouldbe seen as strings from the four dimensional point of view. This realization opens upthe possibility of observing the cosmological consequences of cosmic strings, either fromfundamental strings or from (wrapped) D-branes. It is therefore interesting to understandhow to reconcile the usual quantum mechanical treatment of fundamental strings withtheir expected classical behaviour at cosmological scales. Second, the semiclassical limitdiscussed in this paper may play a role in relation to the possible microscopic counting ofstates [5, 6, 7, 8] associated with known classical supergravity solutions.It is usually assumed that the description of a semiclassical string state (a string ofmacroscopic size) is in terms of a coherent superposition of the fundamental string quanta.However, ealier attempts to build this type of state in the covariant gauge quantizationface serious difficulties [9]. In the next section we motivate the use of the lightconegauge coherent state to give an accurate microscopic description of extended open stringsolutions (spinning string configuration). We contrast the result with the alternative masseigenstates (“perturbative states”) with the same angular momentum, emphasizing theadvantages of the former. In section 3 we focus on the closed string case. We describethe obstacle to the naive extrapolation from the open string case. We then provide thesolution, suitable for the case of a circular target space loop, by gauge-fixing the residualrigid σ translation symmetry. This is done in three ways: in unitary gauge, through aBRST method and in Gupta-Bleuler like quantization. In this section we explore the relation between perturbative quantum states and theirclassical counterparts for open strings. In particular, we will find the closest quantumstate to the leading classical Regge trajectory that corresponds to the following spinningconfiguration, X = Aτ ,X = A sin τ cos σ ,X = A cos τ cos σ . (1)In the conventions of [10], the general solution for the open strings in light-cone gauge isgiven by the following expressions, X + = x + + l p + τ ,X i = x i + l p i τ + il X n =0 n α in e − inτ cos nσ , − = x − + l p − τ + il X n =0 n α − n e − inτ cos nσ . (2)On the other hand, in order to fulfill the constraints the α − n are restricted to be functionsof the physical transverse directions, α − n = 12 lp + ∞ X m = −∞ : α in − m α im : − aδ n ! , (3)where the a coefficcient comes from the normal ordering of the α operators. Furthermorewe can also write the mass and the angular momentum in terms of transverse modes inthe following way, M = 2 l ∞ X n =1 α i − n α in − a ! , (4) J ij = − i ∞ X n =1 n (cid:16) α i − n α jn − α j − n α in (cid:17) . (5) We start out by finding the description in the lightcone gauge of the classical solution ofthe open string found in (1). It is clear that this state corresponds to a string spinningaround its center of mass, which has been set to the origin of the coordinates, whichimplies that, x + = x − = x i = p i = 0 . (6)Also, it is easy to see that the only excited oscillators in this solution are,: α = ( α − ) ∗ = − iA l , (7) α = ( α − ) ∗ = A l . (8)Which in turn implies that, X = il (cid:16) − α − e iτ + α e − iτ (cid:17) cos σ = A cos σ cos τ , (9) X = il (cid:16) − α − e iτ + α e − iτ (cid:17) cos σ = A cos σ sin τ . (10)And for the ( − ) oscillators we obtain, α − = lp − = 12 lp + (cid:18) Al (cid:19) , (11)with all the other α − n = 0. Note that the only non-trivial cases are n = ±
2, which in ourcase are still zero, α − = 12 lp + (cid:16) α α + α α (cid:17) = 12 lp + " − (cid:18) A l (cid:19) + (cid:18) A l (cid:19) = 0 , (12)2nd the same for α −− . If we want the string to move on the 1 − X = 0, X = 1 √ (cid:16) X + − X − (cid:17) = 1 √ l p + − l p − ) τ = 0 , (13)in other words, that p + = p − = A/ √ l . This also means that, X = 1 √ (cid:16) X + + X − (cid:17) = 1 √ l p + + l p − ) τ = Aτ . (14)So, putting all these results together, we finally get a solution of the form (1), X = Aτ ,X = A sin τ cos σ ,X = A cos τ cos σ . (15)We can also use the equations given above to compute, in the classical limit, the observ-ables of this state, i.e. , its mass and angular momentum. M = 2 l ∞ X n =1 α i − n α in ! = 2 l (cid:16) α − α + α − α (cid:17) = A l , (16) J = − i (cid:16) α − α − α − α (cid:17) = A l . (17)which indeed show that these configurations belong to the classical Regge trajectory ofmaximum angular momentum per unit mass. We will now try to obtain the quantum state for the open bosonic string that resemblesthe classical case described above. There seem to be two natural possibilities:
The classical calculation suggests that we construct the quantum state for this solutionin the following way, | ψ i = 1 √ n n ! (cid:16) α − − iα − (cid:17) n | i . (18)The reason to choose this particular configuration becomes clear when we realize that thisstate is, in fact, an eigenstate of mass and angular momentum, M | ψ i = 2 l (cid:16) α − α + α − α − (cid:17) | ψ i = 2 l ( n − | ψ i , (19) J | ψ i = − i (cid:16) α − α − α − α (cid:17) | ψ i = n | ψ i . (20)3sing the identification n = 1 + A / l , we see that this state has identical values of massand angular momentum to the classical configuration in the n ≫ A ≫ l limit. Infact, they saturate the quantum inequality for mass eigenstates given by, J ≤ l M + 1 (21)On the other hand, this is not an eigenstate of the position of the string. It is easy toshow that, in this state, the expectation value of the spatial part of the string positionoperator is equal to zero for all values of σ and τ , namely, h ψ | α in | ψ i = 0 , (22) h ψ | α − k | ψ i = h ψ | lp + ∞ X m = −∞ α ik − m α im | ψ i = 0 , (23)for k = 0. Finally, h ψ | α − | ψ i = n − lp + , (24)which implies that h ψ | X | ψ i = Aτ , (25) h ψ | X i | ψ i = 0 , (26) h ψ | X | ψ i = 0 . (27)We notice that while it is true that this state has similar properties to the classical one,it clearly does not resemble the macroscopic string state, in the sense that its spacetimemotion is not reproduced at all, not even taking a high excitation number ( i.e. , n ≫ On the other hand, it seems more reasonable to try to mimic the classical configurationby constructing a coherent state of the form, | φ i = e vα − − v ∗ α e − ivα − − iv ∗ α | i , (28)where v is a parameter related to p + , ( p + = l − q | v | −
1) and to the amplitude of thespacetime oscillations. Where we have used the relations noted before between the different sets of parameters, p + = A √ l as well as n = A l + 1. h φ | M | φ i = 2 l h φ | ∞ X n =1 α i − n α in − ! | φ i = 2 l h φ | (cid:16) α − α + α − α − (cid:17) | φ i = 2 l (2 | v | − , (29) h φ | J | φ i = − i h φ | (cid:16) α − α − α − α (cid:17) | φ i = 2 | v | , (30)which also correspond to the values obtained in the previous section by considering theidentification 2 | v | = n . The key point, however, is that this state does have the spacetimeposition expectation value of an extended string, namely, h φ | X | φ i = l τ q | v | − A τ , (31) h φ | X | φ i = 2 lv sin τ cos σ = √ A + 2 l sin τ cos σ , (32) h φ | X | φ i = 2 lv cos τ cos σ = √ A + 2 l cos τ cos σ , (33) h φ | X | φ i = 0 , (34)which for A ≫ l approaches the classical solution discussed above. This shows thatthe coherent state is a much closer match to the classical solution than the previouslyconsidered construction.
The calculations in the previous section show how one can obtain a semiclassical coherentstate for open strings in a very similar way to the simple harmonic oscillator. However, nosuch construction is available for closed strings, except in the approach of “semiclassical”quantization (in the sense of [11], where the constraints are satisfied in mean value) asin [12]. In this section we will present the obstacle in finding a macroscopic state fromperturbative closed string states in the lightcone gauge, and propose a solution.
Consider the first order form of of the bosonic string action with world-sheet coordinates m = (0 , ≡ ( σ, τ ), S = 12 πα ′ Z d σ (cid:18) ∂ m X · P m + g mn P m · P n (cid:19) , (35)where g mn = ( − h ) − / h mn is the unit determinant part of the world-sheet metric h mn (related to the second order form [13] using P m = ( − h ) / h mn ∂ n X ). Note that in this limit we also recover the same values for the mass and angular momentum of theclassical configuration. P σ via its equation of motion: P σ = − g ( X ′ + g P τ ) (36)the action becomes S = 12 πα ′ Z d σ " ˙ X · P − g (cid:16) X ′ + P (cid:17) − g g X ′ · P , (37)where P τ ≡ P to simplify notation. The reparametrization symmetry of (35) can be usedto set g = 1 and g = 0. Further, Weyl symmetry can be used to set h = 1 andresidual semilocal symmetry to set X + = τ , P + = 1 (light − cone gauge) . (38)Let us now look closely at the constraint obtained by varying g , i.e. , X ′ · P = 0 thathas the following mode decomposition ( R π dτ e imτ · · · ): C m = α − m − e α − m + X n =0 (cid:16) α im − n α in − e α im − n e α in (cid:17) , (39)where C = ∆ N , (recall α − = e α − ) (40)is the generator of rigid σ shifts ( δX = ǫ∂ σ X , with constant ǫ ). Upon quantization, if aphysical state | phys i satisfies the constraint C :∆ N | phys i = 0 , (41)then, it follows that 0 ≡ h phys | [∆ N, X ] | phys i = ∂ σ h X i . (42)This shows that the string would appear to be stuck at a fixed point h X i for all σ ,making it impossible, in this gauge, to have a macroscopic extended closed string. Thereason for this is that we have not fixed the gauge completely so, in practice, we areintegrating over all the gauges compatible with the lightcone gauge which, of course,yields the aforementioned center of mass position for the whole string. This makes theoperator X not the right quantity to look at in this gauge if we are interested in evaluatingthe semiclassical position of the string. The way out of this problem that we suggest inthe following section is to fix the gauge completely before evaluating the position of thestring. 6 .2 Gauge-fixing σ translations In this subsection we will fix the gauge for the residual rigid reparametrization symmetry σ → σ + ǫ that remains after choosing light-cone gauge for closed strings. We do thisby prescribing the value of one of the coordinates modes of the string. This fixes thesymmetry in a way similar to the way in which the open string arises from the closedstring by removing the modes of one handedness [14].Our goal here will be that of describing a circular string. Once we are in the light-cone gauge, X + = ( X + X ) / √ ∝ τ , we single out two of the coordinates (that spanthe plane in which the circular loop lies): X and X . The solutions for the equationsof motion of these coordinates have the usual decomposition into left and right movingmodes, namely, X i = X iL + X iR . We propose the following additional gauge-fixing condition suitable for the descriptionof the circular string loop states:Φ = 1 π Z dσ e − iσ (cid:16) ∂ − X − vl e − i ( τ − σ ) (cid:17) = 0 , (46)where ∂ − = ∂ τ − ∂ σ and the parameter v will be related to the radius of the circle. Notethat this is only a condition involving the left-moving part of X since ∂ − X R ≡ L gf = λ Φ , (47)such that λ acts as the Lagrange multiplier enforcing the gauge conditions Φ = 0 (i.e.,determining α ).After reaching light-cone gauge, the only remaining piece of the third term in theaction (37) is given by Z dτ g X n (cid:16) α i − n ( τ ) α in ( τ ) − e α i − n ( τ ) e α in ( τ ) (cid:17) , (48)where g stands for the zero mode of g , we have used the decomposition ∂ − X = P α n ( τ )e niσ , . . . etc.Varying with respect to λ and g we obtain the gauge-fixing condition and the con-straint that can be solved (if v = 0) for α ( τ ) and α − ( τ ) respectively. On the solutions, For closed strings the standard decomposition into left and right-movers is X i = X iL + X iR , (43) X iL = 12 x i + 12 l p i ( τ − σ ) + i l X n =0 n α in e − in ( τ − σ ) , (44) X iR = 12 x i + 12 l p i ( τ + σ ) + i l X n =0 n ˜ α in e − in ( τ + σ ) , (45)where i = 1 , · · · ,
8, and x i and p i are the center of mass position and momentum of the loop. n ( τ ) = α n e − inτ , . . . , etc. Therefore, we obtain, α = v , (49) α − = − v X m ≥ α − m α m + X n ≥ α j − n α jn − e α i − n e α in , (50)where j = 2 , · · · ,
8. These results show that the idea behind our gauge fixing choice in (46)is very much like the one used to solve the constraints in the lightcone gauge expressing α − n in terms of the transverse modes.In this gauge, then, the mode decomposition of X L is different from the usual. Itreads, X L = 12 x i + 12 l p i ( τ − σ ) − i l α − e i ( τ − σ ) + i v l e − i ( τ − σ ) + i l X n> n (cid:16) α n e − in ( τ − σ ) − α − n e in ( τ − σ ) (cid:17) , (51)where α − should be interpreted as the operator on the rhs of (50).Alternatively, using the BRST method, the same result can be obtained with a gauge-fixing term linear in Φ. The BRST charge in this case contains an extra term: Q extra = cλ , (52)with corresponding ghost and anti-ghost c and b satisfying { c, b } = 1.The gauge-fixing term in this case is L gf = { Q, Λ } , (53)where Λ = b Φ is the gauge-fixing function. The extra term Q extra gives the contribution(47), and there are also Fadeev-Popov terms from b { Q, Φ } with contribution from theterm originally present in the BRST charge, ˜ c ∆ N . By using the gauge condition thiscontribution is v ˜ cb . Thus, the ghosts decouple in this gauge.Then, one proceeds as before, solving the gauge condition and constraint.Quantization in this gauge can also be achieved in a Gupta-Bleuler approach, byimposing the gauge condition and constraint on physical states. For any pair of physicalstates | χ i and | φ i , we require: h χ | Φ | φ i = 0 . (54)But (54) is satisfied if we impose the following condition on physical states:Φ | phys i = 0 , (55)Therefore, (55) implies, α | phys i = v | phys i , (56)8sing (56), the ∆ N = 0 constraint can be rewritten. Again, at the quantum level, forany pair of physical states | χ i and | φ i h χ | ∆ N | φ i , (57)should hold. But it is enough to impose: vα − + X m ≥ α − m α m + X n ≥ α j − n α jn − e α i − n e α in | phys i = 0 , (58) Let us now consider a state in the gauge of the previous subsection of the following form | φ i = | φ i L ⊗ | n i R , (59)where the left-moving factor is a coherent state built on a left vacuum, | i L , | φ i L = e − ivα − − iv ∗ α | i L , (60)and the right-moving part is an eigenstate of N R with eigenvalue 2 v . We also set theparameters x i and p i to zero, making the center of mass the origin of the coordinatesystem. Notice that α | φ i L = − iv which implies that h φ | α − | φ i L = v upon usingequation (50).We can compute now the expectation value of the string coordinates in the normalizedstate | φ i to find a stationary circular loop of radius vl : h X i = 2 l τ q | v | − , h X i = il h φ | − α − e i ( τ − σ ) + v e − i ( τ − σ ) | φ i , = v l sin 2( τ − σ ) , (61) h X i = il h φ | − α − e i ( τ − σ ) + α e − i ( τ − σ ) | φ i , = v l cos 2( τ − σ ) . (62)There is no contribution from the right-moving bosonic excitations to the expectationvalue because we are considering that this sector is in a N R eigenstate.Before ending this subsection, let us note that the circular loop is just one possiblecoherent state constructed using this gauge. It is not difficult to see the generalization toother shapes. Take a left-moving component for the state of the form: | φ i = e A | i L , A = X { i,m }6 = { , } u mi α i − m + u ∗ mi α im , (63)9 = 1 , · · · , m runs over positive integers. If the right-moving part of the state haslevel N R = n , then h X i is real provided n − v = X { i,m }6 = { , } | u mi | . (64)Choosing the parameters v and n appropriately one can build a loop of arbitrary shapein target space. As a trivial example, consider n = v , u mi = 0 to obtain a folded stringalong the X axis. The use of coherent states allows for the construction of semiclassical states in superstringtheory that bear close resemblance to the classical solutions. These are of relevance inboth studies of some macroscopic defects expected to arise in string theory descriptionsof inflationary cosmology and towards a microscopic entropy counting of certain stringconfigurations that correspond to classical supergravity solutions.For the open string case we have shown that the coherent state reproduces the classicalmotion in the target spacetime while other ’perturbative’ excitations have large oscillations(averaging to zero) around the center of mass position of the string. In the closed stringcase similar properties are found. Also, with the gauge-fixing of section 3 we obtained themicroscopic description of a left-moving static loop supported by a right-moving world-sheet current; the analog of a classical superconducting vorton solution [15].As a final remark let us mention that throughout the text we have used ten dimensionalspacetime thinking about the bosonic part of a superstring, the modification to includethe fermionic sector being straightforward. In particular, the state of subsection 3.3 couldhave fermionic right-moving excitations accounting for part or the whole of the level N R needed. The latter possibility leads to the loop completely stabilized by fermionicexcitations [12] which has no classical gravitational radiation. Acknowledgments
We would like to thank Jaume Garriga, Ken Olum, Alex Vilenkin and especially RobertoEmparan and Jorge Russo for illuminating discussions. AI is grateful to the organizers ofthe Simons Workshop at Stony Brook and Perimeter Institute for their hospitality whilethis work was in progress. The work of AI was supported by DOE Grant DE-FG03-91ER40674. WS was supported in part by NSF Grant PHY-0354776.
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